involves abstruse theorems of the integral calculus. Poincaré's papers, Sylvester's contraction
of Tchebycheff's limits, with reference to the distribution of primes, and researches of J. Hadamard (awarded the
Grand prix of 1892), are among the latest researches in this line. The enumeration of prime numbers has been undertaken at different times by various mathematicians. In 1877 the British Association began the preparation of factor-tables,
under the direction of J. W. L. Glaisher. The printing, by
the Association, of tables for the sixth million marked the completion of tables, to the preparation of which Germany, France, and England contributed, and which enable us to resolve into prime factors every composite number less than .
Miscellaneous contributions to the theory of numbers were made by Cauchy. He showed, for instance, how to find all
the infinite solutions of a homogeneous indeterminate equation of the second degree in three variables when one solution is given. He established the theorem that if two congruences, which have the same modulus, admit of a common solution,
the modulus is a divisor of their resultant. Joseph Liouville
(1809–1882), professor at the Collège de France, investigated mainly questions on the theory of quadratic forms of two, and of a greater number of variables. Profound researches were instituted by Ferdinand Gotthold Eisenstein (1823–1852), of
Berlin. Ternary quadratic forms had been studied somewhat by Gauss, but the extension from two to three indeterminates was the work of Eisenstein who, in his memoir, Neue Theoreme der höheren Arithmetik, defined the ordinal and generic characters of ternary quadratic forms of uneven determinant; and, in case of definite forms, assigned the weight of any order or genus. But he did not publish demonstrations of his results. In inspecting the theory of binary cubic forms, he was led to the discovery of the first covariant ever considered in analysis. He showed that the series of theorems, relating to the presentation of numbers by sums of squares, ceases when the number of squares surpasses eight. Many of the proofs omitted by Eisenstein were supplied by Henry Smith, who
was one of the few Englishmen who devoted themselves to the study of higher arithmetic.
Henry John Stephen Smith[]90 (1826–1883) was born in London, and educated at Rugby and at Balliol College, Oxford. Before 1847 he travelled much in Europe for his health, and at one time attended lectures of Arago in Paris, but after that year he was never absent from Oxford for a single term. In 1861 he was elected Savilian professor of geometry. His first paper on the theory of numbers appeared in 1855. The results of ten years' study of everything published on the theory of numbers are contained in his Reports which appeared in the British Association volumes from 1859 to 1865. These reports are a model of clear and precise exposition and perfection of form. They contain much original matter, but the chief results of his own discoveries were